Availability and irreversibility in thermodynamics

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Availability and irreversibility in thermodynamics
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1951 Br. J. Appl. Phys. 2 183
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ORIGINAL CONTRIBUTIONS
Availability and irreversibility
in
thermodynamics
By
PROFESSOR
JOSEPH
H.
KEENAN,
S.B.,*
Massachusetts Institute of Technology, Cambridge, Mass.,
U.S.A.
[Paper received
22
February,
19511
Connexion is made between Gibbs’s “available energy of the system and medium” and the criterion
of stability. The available energy concept is developed for systems which communicate only with
the uniform medium or atmosphere and for systems which communicate with an additional
reservoir of specified temperature. The treatment
is
extended to problems
of
transient and
steady flow.
A
measure
of
departure from the ideal, the
irreversibility,
is
defined and examined
for its significance. Performance coefficients are devised for several classes of processes. Finally,
the generality of the method
is
exhibited by analysis
of
a variety of thermodynamic phenomena.
INTRODUCTION
The concept of an idealized process with which an actual
process may be compared is common
to
all branches
of
thermodynamics. In view of the second law the idealized
process is usually
so
selected as to be one of maximum
production
of work. The obtaining of maximum work
involves the establishment of some restricting conditions
regarding the possible behaviour of the system. For
example, the physicist and the chemist have sometimes
prescribedan environment of fixed temperaturewithwhich
the otherwise isolated system must be in temperature
equilibrium initially and finally-the maximum work then
being the decrease in the so-called Helmholtz free energy;
or they have sometimes prescribed an environment of k e d
pressure and temperature with which the system must be
in both pressure and temperature equilibrium initially
and finally-the maximum useful work being the decrease
in
the so-called Gibbs free energy; the engineer has some-
times prescribed adiabatic conditions between some
specified initial state and a specified final volume or
pressure-the maximum work being the decrease in
energy or enthalpy (depending upon the nature of the
process) at constant entropy.
In
this latter instance it
has often been pointed out that the final state of the
idealized process may be quite different from that of the
actual process.
None of these idealizations has generality-each is an
ad hoc
device of limited utility. The more general
approach to the statement of the thermodynamically
most beneficial result and to the evaluation
of
departures
from it
has
been given limited attention.
The foundations of the general approach were laid by
J.
W. Gibbs in his second paper
on
thermodynamics.(’)
Gibbs imposed on the behaviour
of
the system the con-
dition that it should be isolated except for communication
with a stable environment of uniform pressure and tem-
perature. He put
no
restrictions, other than those im-
posed by the nature of thermodynamics, upon the
selection
of
the system or of its initial and final states.
Maxwell@ adopted
a
modified version of Gibbs’s
approach which omitted from consideration the pressure
of the environment. Darri e~s,(~) starting from Maxwell’s
*
Temporarily
at
Imperial
College
of
Science
and
Technology,
London.
VOL.
2, JULY 1951
183
method, developed a treatment of engineering problems
in steady
flow
which was expanded upon by Keenad4)
and others. Published quantitative studies of irreversible
processes that stem less directly from the work of Gibbs
are less general in application. For example, the recent
work of Tolman and Fine(’)
is
restricted to cyclic and
steady-flow processes.
The purpose of the present paper is to develop the
method
of
Gibbs more generally than has hitherto been
done. Connexion is made between Gibbs’s “available
energy of the system and medium” and the criterion of
stability. The available energy concept is developed for
systems which communicate only with the uniform
medium or atmosphere and for systems which com-
municate with an additional reservoir of specified tem-
perature. The treatment is extended to problems of
transient and steady flow. A measure
of
departure from
the ideal, the
irreversibility,
is defined and examined for
its significance. Performance coefficients are devised for
several classes of processes. Finally, the generality of
the method is exhibited by analysis of a variety of
thermodynamic phenomena.
In
this generality may be found the justification of the
availability concept. It is through this concept that
processes as widely different as the decay of motion
in
a
viscous fluid, the rectification of
a
binary mixture, and
the dissociation
of
hydrogen peroxide can be examined
from a common basis of comparison and their thermo-
dynamic quality compared quantitatively by means of
the
irreversibility
or
a
coefficient
of
performance
as here
detined.
STABILITY,
MAXIMUM WORK, AND
AVAILABILITY
System exposed
to
an infinite atmosphere
only.-
Virtually all problems which can be treated adequately
by the methods of thermodynamics are terrestrial: that is,
they relate to the behaviour of systems which are sur-
rounded by an essentially infinite atmosphere.
A
major
exception to this latter generalization is found in the
subject of meteorology wherein the system under con-
sideration is the atmosphere itself. For other terrestrial
problems, however, the system considered is small
in
mass and extent compared with the surrounding atmo-
sphere which, for purposes of analysis, may be thought
J.
H,
of
as an environment whose temperature and pressure
are unaltered by any process experienced by the system.
By including within the system as much material or
machinery as is affected by the process (except for the
atmosphere) one may consider any process which occurs
as one in which the system interacts with the atmosphere
only, Gibbda has shown that for any process which can
occur under these circumstances
where
A@
denotes the increase in the quantity
CD
which
is
defined as follows:
where
E
denotes the energy of the system,
V
its volume,
S
its entropy,
po
the pressure of the atmosphere, and
To
its temperature on the Kelvin scale.
Since it is evident that processes or changes can occur
until the pressure of the system is uniformly
po
and its
temperature uniformly
To
and perhaps even thereafter,
then the state from which
no
spontaneous changes can
occur is the state for which the system has the pressurepo
and the temperature
To
and for which
@
has the minimum
of all possible values.
This
is the state
of
stable equi-
librium if there is only one such state, and of neutral
equilibrium and maximum stability if there is more
than one.
A@
9 0
(1)
@
=B+poV-ToS
(2)
Gibbs referred to the difference
@
-
@mi”
where
@
refers to a state in question and
Qmin
to the
most stable state, as the “available energy of the body
and medium” for the state in question(’) (Gibbs uses the
term body for what is called the system here and the term
medium for what is called the
atmosphere
here). By this
he meant the maximum useful work-that
is,
work
in
excess of that done against the atmosphere-which
could be obtained from the system and atmosphere,
without aid from other things, by any possible processes.
The point of view of the last paragraph differs from
that of those which preceded it in that if useful work is
delivered then something other than the system or the
atmosphere must be affected by the process through the
reception of work. The only other thing considered
to
be affected, however, is a work reservoir, such as a coiled
spring or
a
flywheel, which operates adiabatically in the
course of the process.
The proof of Gibbs’s statement, which he does not
give explicitly, may be given as follows. If a system
which is surrounded by an atmosphere at
po
and
To
experiences
a
change from state 1 to state 2 while it
receives net heat @ositive or negative) from the atmo-
sphere only, the useful work which will be delivered to
things other than the system and atmosphere cannot
exceed that of a reversible process between states 1 and
2.
For if this were not true then it would be possible t o
execute the hypothetical process from
1 to
2
which
produces work in excess of that of the reversible process
and to complete a cyclic change for the system by means
Keenan
of the reverse of the reversible process from 2 to 1.
This cycle would have
a
net production of positive work
and, since the atmosphere at
To
would be the only heat
reservoir engaged in the process, the cycle would con-
stitute a perpetual-motion machine of the second kind.
The hypothetical process is therefore impossible and
no
process can produce useful work in excess of that of the
reversible process.
By similar reasoning
it
may be shown that all reversible
processes between states 1 and
2
having heat transfer
between system and atmosphere only must produce
identical quantities of useful work.
In
order
to
evaluate the useful work of an iniinitesimal
reversible process one may set up a reversible means
of
heat transfer between the atmo-
sphere at.
To
and the system at
T,
say, which consists of a reversible
SYSTEM
cyclic enghe of small enough
dimensions
so
that one cyclic
operation will be required t o
absorb or deliver an infinitesimal
amount
of
heat (Fig. 1). Let
dE,
dV, and dS denote respectively
the energy, volume, and entropy
&::v
E
Rs
B
LE
changes experienced by the system
4QOENGtNE
Fig.
1
in
going from the first prescribed
state to the second. These quanti-
ties would obviously have the same values for this change
of state regardless of the nature of the process or the
amount of useful work produced. Let
SQ,
denote the
heat received by the reversible engine at
To
from the
atmosphere. The quantity
SQo
(unlike dV, dE, and dS)
will have different values for the same change of state
in the system, depending upon the nature
of
the process
and the amount of useful work produced. In what
follows the symbols d and
6
will be used to differentiate
between quantities like dV
on
the one hand, which are
k e d by the end states of the system, and
SQo
on
the
other, which are
not so
fixed.
The magnitude of
SQo
may be, of course, either greater
or less than zero.
A
magnitude less than zero would
denote heat flow away from the engine to the atmosphere.
Now the work done by the system and the cyclic engine
in
combination is given by
0
- d E + s Q o
where
dE,
the increase in energy of the system, is also the
increase in energy of the system and the cyclic engine
combined. Of this work the amount podV must be
expended in displacing the atmosphere. Therefore the
useful work of the reversible process which is the maxi-
mum useful work
(SW,),,
of all possible processes is
given by
By the definition of the temperature scale, however,
(8
Wu)max
=
- -
PodV
4-
SQo
where
SQ
denotes the heat received by the system from
184
BRITISH JOURNAL
OF
APPLIED
PHYSICS
Availability and irreversibility
in
thermodynamics
the reversible engine. Moreover, the change of entropy
dS of the system in the course of this reversible process
is given by
dS
=
SQIT
It follows upon substitution into the previous expres-
sion for
(S
W,),, that
(6
WJ,,
=
-
dE
-
PodV
+
TodS
=
-
d@
(3)
SW,
<
-
d@ (4)
and, in general,
An alternative method of deriving (4), which is sug-
gested by the method of Gibbs in proving
(I),
is as
follows. Since the atmosphere is a fluid which is never
in any but stable equilibrium states, one may write
(5)
where subscript a refers t o the atmosphere. From the
second law of thermodynamics it may be said that the
entropy of the system-atmosphere combination cannot
decrease, or that
Gibbs considered thelarge atmosphere as one whose outer
bounds were fixed in position,
so
that
TOSS,
=
SE,
+
poSV,
SS,
+
dS
2
0
( 6)
SV,
+
dV=
0
(7)
Then the useful work of any process between the pre-
scribed end states of the system is given by
SW,=
-SE,-dE
(8)
since the system-atmosphere combination receives net
heat from nothing else.
Upon substituting for
SE,
in
(8)
from
( 5),
eliminating
SS,
and
SV,
by means of (7) and
(8)
respectively, and
noting that
To
can never be less than zero, one gets
which is
(4).
It might be noted that this proof may be modified by
considering the atmosphere
to
be a finite one which is
restrained by an appropriate envelope to pressure
po.
Then instead of (7) and
(8)
one has
S
W,
<
-
dE
-
podV
+
TodS
S
W,
=
-
SE,
-
dE
-
pO(SV,
+
dV)
(9)
Substitution from
(5)
and
(6)
into
(9)
again yields (4).
Under the restrictions imposed-namely, that heat
can be transferred only between system and atmosphere-
useful work can be obtained only from changes of state
for which
@
decreases. Moreover, since the useful work
may be in any degree less than the maximum value, it
may be zero for any change of state for which@ decreases.
Thus spontaneous changes can occur only to states of
less or equal values of
Q.
Changes of state for which
Q
increases can be accom-
plished only with the aid of useful work supplied from
outside the system-atmosphere combination. Such
changes cannot occur spontaneously.
For a finite change from the system in state
1
t o th
system in state 2
2
wu
=
pw,
2
W,,
<
- l d @
or
w,
<
@I
-
@2
(10
It follows that for a given state 1 of the system there is
i
maximum value of the useful work which can be obtainec
from the system-atmosphere combination for all possiblc
changes
of
state of the system. This is, obviously,
@l
-@rni n
where
Qmin
is the minimum value of
Q
for all possible
states of the system. This minimum value corresponds
to
the most stable state of the system
in
the presence
of
the atmosphere. Necessary, but not sufficient, require-
ments for this state are a pressure ofpo and a temperature
of
To.
It is proposed t o call this maximum value of the useful
work corresponding to a state of the system within the
atmosphere the availability instead of the longer term
used by Gibbs and
to
denote it by the symbol
A.
Thus
Availability
=
A
= @
-
@"in
(1
1)
A>O
(12)
It
may be said that for any state of any system in
a
stable atmosphere
and that for the most stable state of the system
R=O
Moreover, for any one system-atmosphere combination,
Q
differs from
A
by a constant. A geometrical representa-
tion of
@
or
A
as the vertical distance above
a
plane in
which two independent properties of the system (such as
V and
S)
were the other two co-ordinates, would be a
surface with its lowest point representing the most stable
state. If two or more points are equally low and lower
than all others, then these are equally stable states and
equilibrium is neutral as between them.
For
the
A
surface,
as
distinguished from the
@
surface, the lowest
point or points will lie at the zero or datum level.
It follows from (1
1)
that the increase in availability for
any process is given by
and from
(IO)
that
4 A=
A@
(14)
W,< - - M= - A@
(1
5)
System exposed
to
a heat reservoir in addition
to
an
infinite atmosphere.-Often one is interested in processes
which involve heat transfer between
a
system and some
reservoir at a temperature different from that of the
atmosphere. As an example of the latter one might cite
the hot reservoir of a power plant.
Of course, the reservoir may be included within the
system for purposes of analysis, and this device yields
a
VOL. 2,
JULY
1951 185
f
J.
H.
Keenan
satisfactory answer
in
terms of change in
@
of the reser-
voir
and the system. For convenience, however, it would
be
better to put the answer in terms of the quantity of
heat withdrawn from the reservoir.
This may be done
as
follows:-
For
a
change of state in the system while it receives
net heat from the atmosphere and from a reservoir
R,
one may, from the point of view of the preceding para-
graph, write from (10)
where the last term refers to the reservoir
R
and the
next-to-the-last to the system as usual. This may be
rewritten in the form
Wu<
- A@- A@R
(16)
which is dQR(TR
-
TO)/TR. The sum is as indicate1
in
(20).
It should be noted that the heat from the reservoir i
treated differently here as compared with the heat fron
the atmosphere. For example, the former appear
explicitly in (19) and
(20),
whereas the latter does not
The implication is that QR, like the change of state of thl
system,
is
prescribed. The heat from the atmosphere
on
the other hand, is not prescribed and varies in fac
with the magnitude of W,. For this reason the symbo
d e R is used for the prescribed infinitesimal quantity
0.
heat from
R
to correspond with the prescribed change:
in properties of the system dE, dV, and dS.
(W,)-
=
-
A@
-
AER - P o ~ ~ R
+
ToASR
(17)
The term reservoir generally implies a system which
passes only through stable states and which if it expands
sphere. Letting
Q R
denote the heat flow from
the
reservoir to the system, one may therefore write
QR
=
-
AER
-
P~AVR
=
-
TRASR
(18)
Substituting from (18) into (17), one gets
TR
--To
FLOW
PROCESSES
AND
MAXI MUM
SHAFT
WORK
Flow across
a
control surface.--(=onsider a closec
exchanges heat Only with the atmosphere.
When the
mass element dm crosses the surface
U
from outside tc
inside, the useful work of the process executed by a
system consisting of all the fluid finally inside
U
is given b j
or contracts does
so
slowly in the presence
of
the atmo-
control surface
U
(Fig.
3)
in a field of fluid flow whid
SW,
Q
-
d@
(19) or
swu
Q
-
@.b'+
(a);
+
4,h)
(21)
Wu
<
-A@
+
e.(,)
TR
-
To
sw,
Q
-
d@
+
de.(,;;-)
(20)
An
alternative proof of (19) and
(20)
may be devised in
which the work of a reversible process is
evaluated for
a
process involving reversible cyclic engines
t o transfer heat across finite temperature differences.
It
will
not suffice, however, merely to insert an engine
betwen the reservoir and the system because then the
heat leaving the reservoir would not be of the same
quantity as that reaching the system. Instead, one may
use two engines (Fig.
2)
:
one to take heat dQR from the
or for an infinitesimal process
where subscript
U
refers to fluid inside surface
U,
super-
scripts
'
and
"
refer to initial and final states, and
4,
denotes the value of
@
per unit mass for fluid outside
o
which is about to enter.
Alternatively
SW,
Q
-
dQU
+
(22)
where d@5 denotes the h"e in the magnitude of
@
summed
UP
for all fluid found within
U
instantaneously.
If
U
lies within an extensive fluid field, then some of the
useful work represented by
(22)
is done on Or by fluid
which follows the element
dm.
The remainder must be
delivered outside Of
U
by a shaft, a piston rod, electrical
conductors, or the like. Calling this part of the useful
work shaft
work
and denoting it by the symbol W,,
one may write
6
w,
=
6
wl4
+
( Px
-
Po)".dm
(23)
The
+
sign in the right-hand member of (23) arises
from the fact that when
( px
-po) is positive and
dm
is
positive (flowing in, by the implied convention) then the
useful work done in pushing
dm
across
U
is negative.
Implied in
(23)
is the simplifying assumption that
only
normal stresses occur at the boundary of dm-that is,
the control surface is
so
placed that no shear exists where
fluid
flows
through the surface. It follows from
(22)
U
dm
Fig.
3
SYSTEM
Fig.
2
and
(23)
that
SWy
<
-
d@u
+
[&
f
( p x
-
po)vX]dm
take heat from the' atmosphere and reject dQR to the
SW,
<
-
d@.,
f
[e.
+
pXvx
-
ToS,]dm
(24)
reservoir and reject heat to the atmosphere and one
to
system.
Now,
since the system draws
on
the atmosphere
Or
only, the maximum useful work for it is simply
-
S@,
where e,,
w,
and
s,
denote the energy, volume, and
and the maximum useful work from the remainder of entropy each per unit mass of the fluid which is about
the
operation is the work of the first reversible engine to cross
U.
186
BRITISH
JOURNAL OF APPLIED PHYSICS
Availability and irreversibility in thermodynamics
If the fluid has appreciable velocity and it flows in a
gravitational field of strength g, then one has
where
U
denotes the energy per unit mass of fluid at rest
at the datum level, c the velocity of the fluid,
z
its heighl
above the datum plane,
g the acceleration of gravity,
and
go
the acceleration given to unit mass by unit force.
Substituting (25) into (24) one gets
or
where
b
G
h
-
To$
and h = u + p v (27)
the latter being the enthalpy.
It
may readily be seen that if a reservoir
R
is considered
to be outside
U
and to supply heat dQR t o the fluid
inside
U
while
dm
flows in, then
SteadyJ70w.-Application to steady flow results in two
important modifications of the more general equations
which were developed above. The first is that
drI,
=
0
where
l-i,
denotes the total value
of
any property, such as
V,
S,
or
@,
of the mass of fluid found within surface
U
at any instant. The second is that the mass
flow
across
a
can be subdivided into two equal mass flows, one in and
one out. The most general of the flow equations
(28)
then becomes
where
Py
denotes the shaft power flowing .out of
a,
Ci,
a summation over all the streams flowing into
U,
CO,,
a summation over all the streams flowing out
of
U,
and dQR/dt the rate of heat flow from reservoir
R
to material inside
a.
This last equation may be com-
pared with the so-called energy equation of steady flow
in the form
where SQ,/dt denotes the rate of heat flow from the
atmosphere
to
material inside
a.
VOL.
2,
JULY
1951
187
IRREVERSIBILITY
A quantitative definition.-From (1 5) a quantitative
definition of irreversibility can be devised. Letting
I
denote the irreversibility of
a
process which the system-
atmosphere combination executes, one may write
I
=
(Wu)mUx
-
wu
(3
1)
=- M- W,,
(32)
= - A@
-
W,, (33)
Substituting into (33) the definition (2) of
@
and for W,
the integrated form of
(S),
and noting from (7) that
one gets
which, in accordance with
(3,
may be reduced to
AV
=
-
AV,
I
=
TOAS
+
aE,
f
P&Va
I
=
T04(Sa
+
S)
(34)
A similar result may be had by using
(9)
without (7).
Thus, the irreversibility becomes equal to the increase
in
entropy of everything involved in the process multiplied
by the temperature of the atmosphere. It is evident from
both (31) and (34) that
I > O
(35)
If heat is received from a reservoir
R
then the irreversi-
bility is given, in accordance with the definition (31), by
Considering the system and atmosphere combined as
a
system, one may write
W"=
-&--E,+
QR
Upon
substituting this into (36) and proceeding as in the
previous paragraph, one gets
I
=
ToA(Sa
+
5'
+
SR)
Irreversibility
in
OW
across a control surface.-
According to (31) one gets for the flow of
dm
across
a
irreversibility of the magnitude
SI
=
(SW,),,
-
6W,
Since (SW,),, differs from
(SW,),,
by the amount
(px
-
po)vxdm and
8
W,, differs from
6
W,
by the same
amount, it follows that
(37)
SI
=
(6
W,),,
-
S
w,
This is
a
consequence, of course, of eliminating by
hypothesis shear stress at the boundary of the mass
dm.
It is not implied, however, that the process of
flow
across
U
is reversible. For example,
a
may coincide with
a thin porous. wall through which the fluid flows irre-
versibly. This would involve shear between the fluid
and the wall, but not between the fluid
dm
and the
surrounding fluid which pushes
it
across
a.
J.
H.
Keenan
It has been shown(@ from the first law of thermo-
per unit mass of a single stream passing in steady flow
across a control surface would be
dynamics that
I
=
TO(S~
-
SI
+
Asp.
+
AS,)
for
a
control surface which is of invariable volume.
Substituting into (37) the expression for
(ST,),,
given
by (28) and that for
S
W,
in (38), one gets
Since dV, is zero by hypothesis, the last equation becomes
SI
=
To(dS,
- sxdm -
(4)
TR
To
or, since the increase in entropy d S of all the fluid finally
found within
U
is
dS
=
dS,
-
sXdm
61
=
To(dS
+
dSR
+
SS,)
(41)
If
the control surface is permitted to change in volume
by amount dVu during this process and
6
W, is made to
include the useful part of the work done at the moving
boundary, or
Cp,
-
p,,)dV,, then the right-hand member
of (38) is altered by adding
a
term -podVC, the right-
hand member of (39) by cancelling out the term
-
podVC, and
(40)
and (41) remain unchanged. Examples
of
this case would be the intake and discharge processes
in
a reciprocating machine.
The equations developed above for flow processes are
applicable to processes involving flow in or out. The
convention adopted requires that for flow in the quantity
dm
should be a positive number, and for flow out a
negative number. In both instances the subscript
x
refers to states just outside the control surface. For flow
in,
the state outside is essentially independent of the
state inside. For flow cut,
on
the other hand, the state
outside may be a direct consequence of a state im-
mediately
on
the other side of the surface
U.
For
example, if no heat ilows across the surface, then the
enthalpy h, will be identical with the enthalpy
h
im-
mediately
on
the other side of
U.
Irreversibility
in
steady
flow.-For steady flow (40)
reduces to
SI=To(-Xsxdm---%)
~ Q R
TR
(42)
where subscripts
1
and 2 refer to entrance and exit
respectively.
An
alternative, though entirely equivalent, develop-
ment of (42) consists of getting the rate of production
of
irreversibility
by subtracting the right-hand member of (30) from that
of (29) and then multiplying through by dt.
COEFFICIENTS OF PERFORMANCE
Various coefficients of performance in the nature of
efficiencies can be devised in view of the first and second
laws
of
thermodynamics. It would seem t o be desirable
t o define these coefficients in such a way that they would
not exceed unity for any processes to which they may
properly be applied.
Starting from the relation
wu <
(WU),,
which is implicit
in
the definitions of its two members,
one can devise the coefficient
G
=
J+'u/(wu)mux
(43)
This coefficient cannot exceed unity, provided that
(
W,),, is greater than zero.
If, however,
(W&,
is less than zero, then W, may
also be less than zero; and since no lower limit can be set
for W, the value of the coefficient may not only exceed
unity, but may actually extend toward infinity. The
coefficient then would be greater in magnitude the greater
the degree of irreversibility.
A
more rational definition
of
a
coeacient where (W,),, is less than zero would be
cz
=
(Wu),,lWu
(44)
a
quantity which under these circumstances increases in
magnitude with decrease in irreversibility and has the
value unity for the reversible process.
By the definition of the irreversibility
1,
(31), one may
write (43) in the form
c1
=
1
-
I/( Wu) m,
(45)
which in combination with (35) indicates as noted above
that
c1
Q
1
1
=
To(-
%,dm
+
dSR
-
SS,)
)
provided that (WJm, is greater than zero.
The
analogous statement for
Cz
is
c2
=
1
-
I/(-Wu)
cz
Q 1
Thus the irreversibility becomes the product of
To
and
the sum of the net flux of entropy out of the control
surface, the increase in entropy for the reservoir
R,
and which indicates that
that for the atmosphere. For example, the irreversibility
BRITISH JOURNAL
OF
APPLIED
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188
Availability and irreversibility in thermodynamics
provided that (Wu),, (and, therefore,
Wu)
is less
than zero.
The coefficient
C1
is appropriate for processes which
are intended t o produce useful work: for example, the
expansion process in
a
heat engine or the combustion
process in
a
Diesel engine. It compares the useful work
produced with the sum of the expenditures of
@
and of
the work-producing capacity of heat from a reservoir;
thus,
The coefficient
C2
is appropriate for processes which
consume useful work: for example, compression com-
bined with heat rejection in a heat pump. It compares
the sum of the gain of availability for the system and
reservoir with the useful work consumed; thus,
Again
c,
Q
1
when
A@
< O
and
c6
<
1
when
A@>
0
Thus
C,
might be used when the principal expenditure
was a loss in availability of
a
system, and when the
principal product was an increase in availability
of
a
system.
The coefficients
C,,
C,,
C3,
and
C,
(but not
Cs
and
CS)
are applicable to cyclic processes in
a
system.
Since for
these, however,
A@
is zero, it follows that
c,
=
c,
TR
-TO
and
c,
=
c,
A@
-
QR(-)
(-
W")
c,
=
TR
(47)
Coefficient
C1
may be used for work-producing cycles,
coefficient
C2
for work-consuming cycles. The use of
CI
in this sensewas implied by Kelvin i n 1848.(') When
C2
is applied to a refrigeration cycle
QR
will be positive and
(TR
-To)
will be negative. For a heat pump
QR
will be
By writing (19) in two other ways two more pairs of
coefficients may be devised. Thus, from
negative and
(TR
-To) will be positive.
For both,
Coefficients can as readily be devised for flow processes
by the method just outlined. One special case has been
proposed and discussed at some length(lO) for steady-flow
processes in turbines in the absence of a reservoir
R.
It derives from (29) and may be stated in the form
(48)
Wu
will be negative.
TR
-To
wU
+
A@
QR(-)
R
one may define
(49)
W,,
+
A@
,,( k 3 0 )
c,
=
TR
(54)
WY
4 1
-
bz)
E =
TR
-To
w,
+A@
where W, here denotes the shaft work produced when
m
units of mass enter the device at the condition corre-
sponding to subscript 1 and an equal mass leaves
at.
a
TR
-To condition corresponding to subscript 2. If terms other
than
b
in (29) are significant, (54) would have to be
modified accordingly.
(50)
Qd,)
and
c,
=
l/C,
=
It is clear that
c3
Q
1
when
QR(+
'
O
and that
c4
Q 1
when
APPLICATIONS
Pure
substance.-A
simple application of some of the
-
- -
equations developed above can be made to a system
that two
independent properties
are
sufscient
to
deter-
mine
the state.
More
specifically, this is a
system
homogeneous in state
in
the absence of capillarity,
motion, gravity, electricity, magnetism, and chemical
For an infinitesimal change of state of unit mass of
Thus
'3
might
be
when
the principal expenditure
consisting
of
a pure substance under conditions such
was heat from a hot reservoir or heat to a cold reservoir,
and
C4
when the principal product was the heating of a
hot reservoir
or
the cooling of a cold one.
From
(51) change.
TR
-
To
Q
-A@
wU
-
Q R (;~ >
one may define
such a system
SW,
Q
-dA
=
-d@
=
-d$
=
-dm
(55)
TR
-
To
wU
-
QR(,->
-A@
(52) where
o
U
+
POW
-
TOS
c,
=
VOL.
2, JULY
1951 189
J. H.
Keenan
as distinguished from the more general quantity
perature.
This
would be true for
a
change of phase
between two saturation states. If,
on
the other hand,
no
such series of intermediate equilibrium states exists
Q
E
+POW
-
Tg
n e property
o
is, for any fixed atmospheric conditions,
a function of two independent properties such as
w
and
J.
Expanding
(55),
one gets
at that Pressure and temperature, then
(58)
is not valid,
because the integration of
(58)
implies the integration
of
(56)
which can be valid only for a path through
equilibrium states. Thus,
(58)
will not hold for a change
6Wu
Q
-
du
-
podv
f
To&
from
a
supersaturated vapour state to a liquid state at
the same pressure and temperature. For such a change
For idnitesimal changes between equilibrium states the (57)
may
be
integrated
over any
path which consists
fist and second laws of thermodynamics give entirely of equilibrium states, such
as
the constant-
- - U=
-Tds$pdv
(56)
temperature path from the supersaturated-vapour state
to saturated vapour to saturated liquid to the liquid
so
that
Upon integration of
(57)
over a path passing through
equilibrium states, one gets for the maximum useful work
the difference between an area
A
on
the pressure-volume
diagram and an area
B
on
the temperature-entropy
diagram (Fig.
4).
(57)
state
(wgfZ,
Fig. 7).
W,
Q
Cp
-
po)dv
-
(T
-
To)&
Fig.
7
The pure substance has been treated by Gibbs at
some length.(“) His discussion omits capillarity, gravity,
electricity, magnetism and chemical change, but includes
motion and non-homogeneity of state. He represents
states by points in a co-ordinate space in which the
vertical axis is the energy axis and the two horizontal
axes are those of entropy and volume. All equilibrium
homogeneous states in this
space
lie
on
a “primitive
in this space and all non-homogeneous equili-
brium states
on
-derived surfaces,” The “availability,”
as defined here, is the vertical distance the state point
lies above a plane which is tangent
to
the primitive
surface at the point representing the most stable state.
Other representations of availability can be devised, but
few observations could be made
from
them that are not
expressed or implied by Gibbs.
Combustion.-Equations (15), (31),
(32),
and
(33)
may
be applied to any thermodynamic system that may be
considered t o be enclosed within
a stable and uniform
atmosphere. They may, therefore, be applied to processes
involving, mixing, solution, or chemical reaction.
In
order to determine the change in
@
or
A
and the value
of (Wu)ma it is necessary only to know the change
in
P
V
S
Fig.
4
In Particular, for C o Ob
at
constant-volume
(h
<
0)
the maximum useful work is the work which would be
obtained from a reversible cyclic engine which receives
heat from the System at the variable temperature
T
and
rejects
heat to
the
atmosphere
at
the
constant
perature To. This is the area
B’
of Fig.
5.
For expansion
at constant
entropy
the
maximum
usefu1 work the
expansion work in excess of that done against the atmo-
sphere in a reversible adiabatic change. This
is
the area
A’ in Fig.
6.
S=CONSTANT
-
--
P
=
Po
S
V
energy, volume, and entropy of all the material which is
Fig.
5
Fig. 6
involved in the process.
serve as an illustration.
A highly simplified case of a chemical reaction will
Consider a reaction like the formation of nitric oxide,
Between two states having the same pressure and
temperature
(57)
may be integrated to give
W,
Q (p
-po)Av
-
(T
-
To)As
(58)
+Nz
+
$0,
=
NO
provided that a continuous series of intermediate for which the number of moles of products is the same
equilibrium states exist at the same pressure and tem- as the number of moles of reactants and for which both
190
BRITISH
JOURNAL
OF
APPLIED PHYSICS
Availability
and
irreversibility in thermodynamics
reactants and products are, say, diatomic. The problem
may be further simplified by the assumption that reactants
and products are perfect gases having identical molal
specific heats which are independent of temperature.
For an adiabatic reaction at constant volume the
energy of the system is unaltered. Letting subscripts
P
and
R
refer t o products and reactants respectively,
one may write
In
terms of the energy of reaction
u p
-
=
0
(59)
where
CO
is the specific heat at constant volume of the
system either as products or as reactants, its value being
by hypothesis the same constant for both. From (61) it
is clear that the temperature rise in combustion
(Tp
-
TR)
is constant and, therefore, independent of the tempera-
ture
TR
at which the reaction begins.
The maximum useful work for this process-or the
decrease
in
availability-is given by
and
W, is in
this case zero; therefore,
I =
(Wu)m
and the irreversibility is seen to decrease with increase
in
the temperature at which the reaction begins. This
is
a
major element in the justification for compression
of a
charge in an internal-combustion engine before ignition.
If the combustion occurs adiabatically at constant
pressure
p,
instead of at constant volume, the result is
similar. Once more the temperature rise is independent
of the temperature of the reactants. The maximum
useful work becomes
(W,>mu
=
@
-
P0) FP
-
VR)
-k
TO(SP
-
SR)
This time, however, the useful work is not zero but
W,
=
@
-
p0)FP
-
vR)
and the irreversibility is, as before,
I
=
To(Sp
-
S,)
which again is a quantity which decreases with increase
in the temperature at which reaction begins.
Flow into a chamber.-The adiabatic flow of a perfect
gas from
a
region of constant conditions into an evacuated
chamber may be used to illustrate a problem in
non-
steady 00w. Referring to conditions in the outside
region by subscript
x
one may write for (38)
and for (28)
SW,=O=
-dE,+h,dm
(65)
(SW,),,
=
-do,
+
b,dm
because both
(Up
-
UR)
and
(Vp
-
VR) are zero.
But Upon integrating each of these between the initial con-
dition of
no
fluid inside the chamber to the final condition,
for which subscriptfwill be used to denote the condition
in the chamber, one gets
(SP
-
Sd
=
(Sp
-
SPO)
-
(SR
-
SRO)
+
( S ~ p ) ~
TP
=
Cv(1n
To
TP
=
C,
In
-
f
(SRP)O
T R
-
In
‘“>
+
(sRp)O
-muf+
mh,
=
0
(67)
(63)
and
(8W,,),u
=
-my5f+
Qj
+
mb,
(68)
where
(SRP)o
is the entropy of reaction at
TO.
stituting (63) into (62), one gets
Sub-
Since, from (61),
( uRP) O
- I - -
T P
TR
TR
c,
the temperature ratio
Tp/TR
decreases with increase in
temperature (for
URp
less than zero-that is, for a
temperature rise in combustion).
It
follows from (64)
that the maximum useful work decreases with increase
in the temperature of the reactants.
The irreversibility is defined by
_ -
where
m
denotes the mass which flows into the chamber.
It is necessary to introduce
ai,
the initial value of
0
in
the chamber, into (68) because, unlike
Vi,
it is not zero.
Specifically,
@j
=
uj
i
povj
-
TOSj
=
P o 6
=
Pomvf
From (67) one gets
U/=
h,
or Tf
=
kT,
where
k
=
C,/c,
from (68) and (69)
J.
H.
w>*.
IC
\ien-
of
c@).
reduces
to
(i*;b,
172
=
ZdSj-
Sx)
=
To(cp In
k
-
R
lnpf/pJ
(72)
F.5
xmi:
aul d
hsve been anticipated by noting that the
r.-m--.
:...,...
.
.
-.
sksfi
w r k per unit of mass entering is also the
k v ~ x 3 i I i r y
of
~s
proclss and is therefore the product
G~T:
.-d
b e
i ncrme in entropy of the fluid.
1:
-...
.&.&y
be
noted
from (72) that the maximum shaft
n.,-k
L--
..-
;id
the irreversibility per unit mass are infinite for
&e
5s
elementary mass which enters the chamber
c~
=
01
md that
just
before flow ceases
(pf
=
px)
these
~x=t i &s
s e reduced to c,T, In
k.
T?<
s.’cam
poii.er-pIant
cycle.-A steady-flow steam
pn-er-pIam cycle may be used to illustrate several
q p
of
application of the availability, maximum work,
and irrelersibility concepts. The various processes
invoh-ed may be classified according to whether they are
(e)
adiabatic-that is, employing
no
heat reservoirs
enema1 to the water passing through the cycle,
(b)
ex-
chan@ng heat with the atmosphere only, and (c) exchang-
ing heat with a hot reservoir.
Examples of
(a)
are expansion in a turbine, compression
in a feed pump, and heating feed water with steam bled
from the turbine. An example
of
(b)
is the condensation
process. Examples of
(c)
are the steam-generator
processes
:
namely, heating of feed water, evaporation,
superheating, and reheating.
The change in availability and the maximum useful
work (W,) of a unit mass of fluid may be evaluated for
each process in the cycle. These are, however, of less
engineering interest than the maximum shaft work
(W,),,
for each process. For the cycle as a whole the
summations
of
(W,),, and (W,),, are identical
because their difference, the useful work done on adjacent
fluid, sums up t o zero. The irreversibility summed up
for
the cycle will be equal to the difference between the
summation
of
the maximum shaft work (or maximum
useful work) and the net work of the cycle.
In dealing with the
(c)
type of process-namely, that
involving heat exchange with a hot reservoir-several
choices are open to the analyst. He may, for example,
choose to employ as
a
reservoir a stream of products of
combustion such as might be found
in
an actual power
plant.
In
that event (29), with the last term equal to zero,
must be applied to the stream of water and to the stream
of
products of combustion as well in order to find the
maximum shaft work. This method brings into con-
sideration the characteristics and behaviour of the stream
of products of combustion. A logical extension of it
would be
to
include in the application of
(29)
the entire
process of the air-fuel stream from its entrance to the
power plant in the form of reactants
to
its exit in the form
of
cooled products of combustion. Any analysis which
includes the behaviour of the stream of hot gases is, of
course, concerned with a non-cyclic process.
It
might be desired, however, to study a cyclic process
Keenan
as such in which heat exchange is with conventional
“reservoirs.” In that event, the temperature of the hot
reservoir is arbitrary. For a given cycle the irreversi-
bility will be greater the higher the temperature selected
for the hot reservoir, and the lowest magnitude of the
irreversibility will correspond to the lowest possible
temperature of the reservoir.
For
example, in
a
steam
cycle the lowest possible temperature for a single hot
reservoir which supplies heat to the cycle without the aid
of intermediate cyclic machinery would be the highest
temperature attained by the steamin the course
of
heating.
If a reservoir temperature is selected higher than this
minimum, the increase in the value of the irreversibility
will be equal
to
the maximum useful work which could
be obtained by virtue of heat transfer from the higher
temperature and to the lower one-the magnitude of the
heat being that taken from either reservoir by the cycle.
Thus the increase in irreversibility is attributable to heat
flow across the finite difference between the temperatures
of the two reservoirs.
CONCLUSI ONS
Quantitative concepts of maximum useful work,
availability, irreversibility, and quality of performance of
a thermodynamic task may be defined from considera-
tion of the first and second laws of thermodynamics for
all processes between equilibrium states of a system
operating within an infinite stable atmosphere. These
concepts may be extended to cover flow across a control
surface and, as
a
more special case, to steady flow through
a control surface. They may be applied t o as wide a
range of processes and as great
a
variety of systems as
the science of thermodynamics itself.
REFERENCES
(1)
The Collected Works
of
J.
Willard Gibbs,
p.
39 (London:
Longmans, Green and Co. Ltd., 1931).
(2)
MAXWELL,
J.
C.
Theory
of
Heat,
10th edition,
p.
195
(London: Longmans, Green and
Co.
Ltd., 1891).
(3)
DARRIEUS. Rev. Gen. de I’Elect.,
27,
p.
963 (1930); see
also
Engineering,
130, p.
283 (1930).
(4)
KEENAN,
J.
H.
Mech. Engng,
54,
p.
195 (1932);
Thermodynamics
(New York:
John
Wiley and Sons
Inc., 1941).
( 5)
TOLMAN
and
FINE.
Rev.
Mod.
Phys.,
20,
p.
51 (1948).
( 6) The Collected Works
of
J.
Willard Gibbs,
1,
p.
40
(London: Longmans, Green and
Co.
Ltd., 1931).
(7)
The Collected Works
of
J.
Willard Gibbs,
1,
p.
53
(London: Longmans, Green and Co. Ltd., 1931).
(8) KEENAN,
J.
H.
Thermodynamics,
p.
34 (New York:
John
Wiley and Sons Inc., 1941).
(9)
THOMSON,
SIR WILLIAM. Trans. Roy. Soc. Edinb.,
16,
(1849); see
also
Math. and Phys. Papers,
p.
152
(London: Cambridge University Press, 1882).
KEENAX,
J.
H.
Mech. Engng,
54, p.
195
(1932).
(10)
DARRIEUS. Engineering,
130,
p.
283 (1930).
(1 1)
The Collected Works
of
J.
Willard Gibbs,
p.
33
(London:
Longmans, Green and Co. Ltd.,
1931).
192
BRITISH
JOURNAL
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APPLIED
PHYSICS